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Theorem eldprdi 19259
Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
eldprdi.0 0 = (0g𝐺)
eldprdi.w 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
eldprdi.1 (𝜑𝐺dom DProd 𝑆)
eldprdi.2 (𝜑 → dom 𝑆 = 𝐼)
eldprdi.3 (𝜑𝐹𝑊)
Assertion
Ref Expression
eldprdi (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))
Distinct variable groups:   ,𝐹   ,𝑖,𝐺   ,𝐼,𝑖   0 ,   𝑆,,𝑖
Allowed substitution hints:   𝜑(,𝑖)   𝐹(𝑖)   𝑊(,𝑖)   0 (𝑖)

Proof of Theorem eldprdi
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eldprdi.1 . 2 (𝜑𝐺dom DProd 𝑆)
2 eldprdi.3 . . 3 (𝜑𝐹𝑊)
3 eqid 2738 . . 3 (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)
4 oveq2 7178 . . . 4 (𝑓 = 𝐹 → (𝐺 Σg 𝑓) = (𝐺 Σg 𝐹))
54rspceeqv 3541 . . 3 ((𝐹𝑊 ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)) → ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
62, 3, 5sylancl 589 . 2 (𝜑 → ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
7 eldprdi.2 . . 3 (𝜑 → dom 𝑆 = 𝐼)
8 eldprdi.0 . . . 4 0 = (0g𝐺)
9 eldprdi.w . . . 4 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
108, 9eldprd 19245 . . 3 (dom 𝑆 = 𝐼 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
117, 10syl 17 . 2 (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
121, 6, 11mpbir2and 713 1 (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wrex 3054  {crab 3057   class class class wbr 5030  dom cdm 5525  cfv 6339  (class class class)co 7170  Xcixp 8507   finSupp cfsupp 8906  0gc0g 16816   Σg cgsu 16817   DProd cdprd 19234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-ixp 8508  df-dprd 19236
This theorem is referenced by:  dprdfsub  19262  dprdf11  19264  dprdsubg  19265  dprdub  19266  dpjidcl  19299
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